How do I find out the name of ships that docked at Whitby Harbour during June, 1840?

The best place to start looking for possible records in this instance is by contacting the harbour master at Whitby and asking if log books were kept at that time and if so where they would be located. A lot of harbours have marine and historical societies where one might have access to archives including log books. Contact information for these types of lake front societies may be available at Great Lakes Shipping Genealogy Downward Bound {Special Thank you to Laura Carter for her help in answering this question}

Try the municipal offices of the town of Whitby, or try the Department of Transportation if they have a "Great Lakes" division. Maybe even Archives Ontario under shipping?